In this section, we establish sufficient conditions for the equilibrium equation Eq.(REF ) to be well-posed in the sense that the existence of the equilibrium point \(Z\)  is uniquely determined. Provided the existence of the equilibrium point \(Z\) , we can derive the gradients of the parameters by using the implicit function theorem instead of back-propagating all intermediate layers.
Previous work [1]} has shown the existence of the unique equilibrium point in the vector-based equilibrium equation Eq. (REF ) if the scalar \(\gamma > 0\)  is chosen small enough. We extend this result to a general matrix-based mapping.
